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    Existence of solution legendre equation

    Hi all, I have my exam in differential equations in one week so I will probably post alot of question. I hope you wont get tired of me! Homework Statement This is Legendres differential equation of order n. Determine an interval [0 t_0] such that the basic existence theorem guarantees a...
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    Show that limit set of dynamical system is closed

    Homework Statement Define the w-limit set (omega) of a point. Show that w(x) is closed. Homework Equations The Attempt at a Solution The definition of a limit set is the set of points to which there exists a sequence t_n→∞ such that \phi(t_n,x) → y The second question. I was...
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    Distance between point and set

    \inHomework Statement Denote by d(x,A) = inf |x-y|,y \in A, the distance between a point x \in R^n and a set A \subseteq R^n. Show |d(x,A)-d(z,A)| \leq |x-z| In particular, x → d(x,A) is continuous Homework Equations The Attempt at a Solution I have no idea on how to prove this. I drew...
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    Hard fourier transform

    Homework Statement Using that the Fourier transform of e^{|x|} is \frac{2}{\xi^2+1}. Compute the Fourier transform of \frac{x}{(x^2+1)^2} Homework Equations The Attempt at a Solution My first thought was to try and rewrite the problem in a form I recognized, tried a couple of things but...
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    Fourier transform

    Homework Statement Hi y'all, ran into some trouble with a fourier transform Im supposed to find the fourier transform of f(x)=\frac{1}{x^{2}+6x+13} Homework Equations Not that I know The Attempt at a Solution I tried integrating this with no luck. All help is as usual...
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    Minimizing a integral

    Homework Statement Determine the polynomial of the form P(x)=x^3+ax^2+bx+c that minimizes \int[P(x)]^2 Homework Equations The Attempt at a Solution My first thought was that I should find a second degree polynomial that minimizes x^3. That didn't work at all! So now I'm...
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    PDE: heat equation

    Homework Statement u_{t}=3u_{xx} x=[0,pi] u(0,t)=u(pi,t)=0 u(x,0)=sinx*cos4x Homework Equations The Attempt at a Solution with separation of variables and boundry conditions I get: u(x,t)= \sumB_{n}e^-3n^{2)}}*sinnx u(x,0)=sinx*cos4x f(x)=sinx*cos4x=\sumB_{n}*sinnx...
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    Hard fourier series

    Homework Statement Let f be the 2pi periodic function defined by f(x)=e^{cos(x^{2})} for 0 < x < 2pi. What is the value of the Fourier series at x=4pi Homework Equations The Attempt at a Solution I don't even know where to start. All help is much appreciated!
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    Inital value problem, Laplace transform

    Homework Statement Solve inital value problem: x''+2x'+x=g(t) g(t)= t 0<t<1 2-t 1<t<2 0 t>2 Homework Equations Second shift theorem, Heaviside function and Laplace transforms. I denote Heaviside,functuon H(t-a), and Laplace transform with L The Attempt at a Solution I...
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    Double integral, polar coordinates

    Homework Statement Evaluate \int\intT (x^2+y^2) dA, where T is the triangle with the vertices (0,0)(1,0)(1,1) Homework Equations The Attempt at a Solution \int d\theta \int r^3 dr Thats how far I got, not really sure about boundries on r. First integrals boundrie should be 0 to pi/4. Is...
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    Finding volumes of a solid

    This shouldn't be a problem but I just can't this right! Homework Statement Find the volume of the body that is generated when y= 1/x(sqrt(4-x^2)) ,1 <= x <2, is rotated around the y-axis Homework Equations Using cylindrical shells : 2 pi ∫x (g(x)-f(x)) dx The Attempt at a Solution 2 pi...
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